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Chapter 3 Text Real Analog - Circuits 1 Chapter 3: Nodal and Mesh Analysis 3. Introduction and Chapter Objectives In Chapters 1 and 2, we introduced several tools used in circuit analysis: Ohm’s law, Kirchoff’s laws, and circuit reduction Circuit reduction, it should be noted, is not fundamentally different from direct application of Ohm’s and Kirchoff’s laws – it is simply a convenient re-statement of these laws for specific combinations of circuit elements. In Chapter 1, we saw that direct application of Ohm’s law and Kirchoff’s laws to a specific circuit using the exhaustive method often results in a large number of unknowns – even if the circuit is relatively simple. A correspondingly large number of equations must be solved to determine these unknowns. Circuit reduction allows us, in some cases, to simplify the circuit to reduce the number of unknowns in the system. Unfortunately, not all circuits are reducible and even analysis of circuits that are reducible depends upon the engineer “noticing” certain resistance combinations and combining them appropriately. In cases where circuit reduction is not feasible, approaches are still available to reduce the total number of unknowns in the system. Nodal analysis and mesh analysis are two of these. Nodal and mesh analysis approaches still rely upon application of Ohm’s law and Kirchoff’s laws – we are just applying these laws in a very specific way in order to simplify the analysis of the circuit. One attractive aspect of nodal and mesh analysis is that the approaches are relatively rigorous – we are assured of identifying a reduced set of variables, if we apply the analysis rules correctly. Nodal and mesh analysis are also more general than circuit reduction methods – virtually any circuit can be analyzed using nodal or mesh analysis. Since nodal and mesh analysis approaches are fairly closely related, section 3.1 introduces the basic ideas and terminology associated with both approaches. Section 3.2 provides details of nodal analysis, and mesh analysis is presented in section 3.3. After completing this chapter, you should be able to: Use nodal analysis techniques to analyze electrical circuits Use mesh analysis techniques to analyze electrical circuits © 2012 Digilent, Inc. 1 Real Analog – Circuits 1 Chapter 3.1: Introduction and Terminology 3.1: Introduction and Terminology As noted in the introduction, both nodal and mesh analysis involve identification of a “minimum” number of unknowns, which completely describe the circuit behavior. That is, the unknowns themselves may not directly provide the parameter of interest, but any voltage or current in the circuit can be determined from these unknowns. In nodal analysis, the unknowns are the node voltages. In mesh analysis, the unknowns are the mesh currents. We introduce the concept of these unknowns via an example below. Consider the circuit shown in Figure 3.1(a). The circuit nodes are labeled in Figure 3.1(a), for later convenience. The circuit is not readily analyzed by circuit reduction methods. If the exhaustive approach toward applying KCL and KVL is taken, the circuit has 10 unknowns (the voltages and currents of each of the five resistors), as shown in Figure 3.1(b). Ten circuit equations must be written to solve for the ten unknowns. Nodal analysis and mesh analysis provide approaches for defining a reduced number of unknowns and solving for these unknowns. If desired, any other desired circuit parameters can subsequently be determined from the reduced set of unknowns. (a) Circuit schematic (b) Complete set of unknowns Figure 3.1. Non-reducible circuit. In nodal analysis, the unknowns will be node voltages. Node voltages, in this context, are the independent voltages in the circuit. It will be seen later that the circuit of Figure 3.1 contains only two independent voltages – the voltages at nodes b and c1. Only two equations need be written and solved to determine these voltages! Any other circuit parameters can be determined from these two voltages. Basic Idea: In nodal analysis, Kirchoff’s current law is written at each independent voltage node; Ohm’s law is used to write the currents in terms of the node voltages in the circuit. 1 The voltages at nodes a and d are not independent; the voltage source VS constrains the voltage at node a relative to the voltage at node d (KVL around the leftmost loop indicates that vab = VS). © 2012 Digilent, Inc. 2 Real Analog – Circuits 1 Chapter 3.1: Introduction and Terminology In mesh analysis, the unknowns will be mesh currents. Mesh currents are defined only for planar circuits; planar circuits are circuits which can be drawn in a single plane such that no elements overlap one another. When a circuit is drawn in a single plane, the circuit will be divided into a number of distinct areas; the boundary of each area is a mesh of the circuit. A mesh current is the current flowing around a mesh of the circuit. The circuit of Figure 3.1 has three meshes: 1. The mesh bounded by VS, node a, and node d 2. the mesh bounded by node a, node c, and node b 3. the mesh bounded by node b, node c, and node d These three meshes are illustrated schematically in Figure 3.2. Thus, in a mesh analysis of the circuit of Figure 3.1, three equations must be solved in three unknowns (the mesh currents). Any other desired circuit parameters can be determined from the mesh currents. Basic Idea: In mesh analysis, Kirchoff’s voltage law is written around each mesh loop; Ohm’s law is used to write the voltages in terms of the mesh currents in the circuit. Since KVL is written around closed loops in the circuit, mesh analysis is sometimes known as loop analysis. Figure 3.2. Meshes for circuit of Figure 3.1. © 2012 Digilent, Inc. 3 Real Analog – Circuits 1 Chapter 3.1: Introduction and Terminology Section Summary: In nodal analysis: a. Unknowns in the analysis are called the node voltages b. Node voltages are the voltages at the independent nodes in the circuit c. Two nodes connected by a voltage source are not independent. The voltage source constrains the voltages at the nodes relative to one another. A node which is not independent is also called dependent. In mesh analysis: a. Unknowns in the analysis are called mesh currents. b. Mesh currents are defined as flowing through the circuit elements which form the perimeter of the circuit meshes. A mesh is any enclosed, non-overlapping region in the circuit (when the circuit schematic is drawn on a piece of paper. Exercises: 1. The circuit below has three nodes, A, B, and C. Which two nodes are dependent? Why? 2. Identify meshes in the circuit below. © 2012 Digilent, Inc. 4 Real Analog – Circuits 1 Chapter 3.2: Nodal Analysis 3.2: Nodal Analysis As noted in section 3.1, in nodal analysis we will define a set of node voltages and use Ohm’s law to write Kirchoff’s current law in terms of these voltages. The resulting set of equations can be solved to determine the node voltages; any other circuit parameters (e.g. currents) can be determined from these voltages. The steps used to in nodal analysis are provided below. The steps are illustrated in terms of the circuit of Figure 3.3. Figure 3.3. Example circuit. Step 1: Define reference voltage One node will be arbitrarily selected as a reference node or datum node. The voltages of all other nodes in the circuit will be defined to be relative to the voltage of this node. Thus, for convenience, it will be assumed that the reference node voltage is zero volts. It should be emphasized that this definition is arbitrary – since voltages are actually potential differences, choosing the reference voltage as zero is primarily a convenience. For our example circuit, we will choose node d as our reference node and define the voltage at this node to be 0V, as shown in Figure 3.4. © 2012 Digilent, Inc. 5 Real Analog – Circuits 1 Chapter 3.2: Nodal Analysis Figure 3.4. Definition of reference node and reference voltage. Step 2: Determine independent nodes We now define the voltages at the independent nodes. These voltages will be the unknowns in our circuit equations. In order to define independent nodes: “Short-circuit” all voltage sources “Open-circuit” all current sources After removal of the sources, the remaining nodes (with the exception of the reference node) are defined as independent nodes. (The nodes which were removed in this process are dependent nodes. The voltages at these nodes are sometimes said to be constrained.) Label the voltages at these nodes – they are the unknowns for which we will solve. For our example circuit of Figure 1, removal of the voltage source (replacing it with a short circuit) results in nodes remaining only at nodes b and c. This is illustrated in Figure 3.5. Figure 3.5. Independent voltages Vb and Vc. © 2012 Digilent, Inc. 6 Real Analog – Circuits 1 Chapter 3.2: Nodal Analysis Step 3: Replace sources in the circuit and identify constrained voltages With the independent voltages defined as in step 2, replace the sources and define the voltages at the dependent nodes in terms of the independent voltages and the known voltage differences. For our example, the voltage at node a can be written as a known voltage Vs above the reference voltage, as shown in Figure 3.6. Figure 3.6. Dependent voltages defined. Step 4: Apply KCL at independent nodes Define currents and write Kirchoff’s current law at all independent nodes.
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